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A university is interested in analysing the values of research grants awarded to members of its academic staff. It believes that the total has increased over the years. Assume that X, the value received in research grants in a given year, follows a Normal distribution, with mean μ (in thousands of £) and variance o². Two separate random samples of nf female academic staff and nm male academic staff were obtained for this study. Consider that n = nf + nm and that nm < nf. Let the estimators based on the different samples be nm Σm Xi XF: ΣΧ nf and XM= nm We can define a combination of these two estimators as nm X= XM. nf nf + nm -XF+ nf + nm (a) Show that these three estimators are unbiased for u and that Var(X) < Var(XF) < Var(Xm). (b) Which of these three estimators for u would you recommend? Use the result in (a) to defend your choice. (c) Suppose that the value obtained for X was 52, with n = 30, and o is known to be equal to 4. If we assume μ = 50, what is the probability of observing a sample mean larger than the value obtained? =

A university is interested in analysing the values of research grants awarded to members of its academic staff. It believes that the total has increased over the years. Assume that X, the value received in research grants in a given year, follows a Normal distribution, with mean μ (in thousands of £) and variance o². Two separate random samples of nf female academic staff and nm male academic staff were obtained for this study. Consider that n = nf + nm and that nm < nf. Let the estimators based on the different samples be nm Σm Xi XF: ΣΧ nf and XM= nm We can define a combination of these two estimators as nm X= XM. nf nf + nm -XF+ nf + nm (a) Show that these three estimators are unbiased for u and that Var(X) < Var(XF) < Var(Xm). (b) Which of these three estimators for u would you recommend? Use the result in (a) to defend your choice. (c) Suppose that the value obtained for X was 52, with n = 30, and o is known to be equal to 4. If we assume μ = 50, what is the probability of observing a sample mean larger than the value obtained? =

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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A university is interested in analysing the values of research grants awarded to members of its academic staff. It believes that the total has increased over the years. Assume that X, the value received in research grants in a given year, follows a Normal distribution, with mean µ (in thousands of £) and variance o². Two separate random samples of nf female academic staff and nm male academic staff were obtained for this study. Consider that n = nƒ+nm and that nm < nf. Let the estimators based on the different samples be nm anf Σm Xi ΣΧ nf XF= and Хм nm We can define a combination of these two estimators as nm nf X -XF+ XM. + Nm nf nf + Nm (a) Show that these three estimators are unbiased for u and that Var(X) < Var(XF) < Var(Xµ). (b) Which of these three estimators for u would you recommend? Use the result in (a) to defend your choice. (c) Suppose that the value obtained for X was 52, with n = 30, and o is known to be equal to 4. If we assume µ = 50, what is the probability of observing a sample mean larger than the value obtained? (d) The university believes that the mean should be larger than 50, given their efforts to increase this value. The answer in (c) could be used to test the university's hypothesis. Explain how. -
Transcribed Image Text:A university is interested in analysing the values of research grants awarded to members of its academic staff. It believes that the total has increased over the years. Assume that X, the value received in research grants in a given year, follows a Normal distribution, with mean µ (in thousands of £) and variance o². Two separate random samples of nf female academic staff and nm male academic staff were obtained for this study. Consider that n = nƒ+nm and that nm < nf. Let the estimators based on the different samples be nm anf Σm Xi ΣΧ nf XF= and Хм nm We can define a combination of these two estimators as nm nf X -XF+ XM. + Nm nf nf + Nm (a) Show that these three estimators are unbiased for u and that Var(X) < Var(XF) < Var(Xµ). (b) Which of these three estimators for u would you recommend? Use the result in (a) to defend your choice. (c) Suppose that the value obtained for X was 52, with n = 30, and o is known to be equal to 4. If we assume µ = 50, what is the probability of observing a sample mean larger than the value obtained? (d) The university believes that the mean should be larger than 50, given their efforts to increase this value. The answer in (c) could be used to test the university's hypothesis. Explain how. -
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